Real zeros of random trigonometric polynomials with dependent coefficients

We further investigate the relations between the large degree asymptotics of the number of real zeros of random trigonometric polynomials with dependent coefficients and the underlying correlation function. We consider trigonometric polynomials of the form $$f_n(t):= \frac{1}{\sqrt{n}}\sum_{k=1}^{n}a_k \cos(kt)+b_k\sin(kt), ~x\in [0,2\pi],$$ where the sequences $(a_k)_{k\geq 1}$ and $(b_k)_{k\geq 1}$ are two independent copies of a stationary Gaussian process centered with variance one and correlation function $\rho$ with associated spectral measure $\mu_{\rho}$. We focus here on the case where $\mu_{\rho}$ is not purely singular and we denote by $\psi_{\rho}$ its density component with respect to the Lebesgue measure $\lambda$. Quite surprisingly, we show that the asymptotics of the number of real zeros $\mathcal{N}(f_n,[0,2\pi])$ of $f_n$ in $[0,2\pi]$ is not related to the decay of the correlation function $\rho$ but instead to the Lebesgue measure of the vanishing locus of $\psi_{\rho}$. Namely, assuming that $\psi_{\rho}$ is $\mathcal{C}^1$ with H\"older derivative on an open set of full measure, one establishes that $$\lim_{n \to +\infty} \frac{\mathbb E\left[\mathcal{N}(f_n,[0,2\pi])\right]}{n}= \frac{\lambda(\{\psi_{\rho}=0\})}{\pi \sqrt{2}} + \frac{2\pi - \lambda(\{\psi_{\rho}=0\})}{\pi\sqrt{3}}.$$ On the other hand, assuming a sole log-integrability condition on $\psi_{\rho}$, which implies that it is positive almost everywhere, we recover the asymptotics of the independent case, i.e. the limit is $\frac{2}{\sqrt{3}}$. Besides, with further assumptions of regularity and existence of negative moment for $\psi_{\rho}$, we moreover show that the above convergence in expectation can be strengthened to an almost sure convergence.